Mirror, Mirror

The point P(5, 3) was reflected over a line in the x,y plane. Its resulting position was P' at (-3, 9). The line it was reflected over can be written in standard form as Ax + By = C. A, B, and C are integers with no common factors, and A is a positive integer. Find A + B + C.

The line of reflection is identical to the perpendicular bisector of the line segment between corresponding points on the preimage and image of a reflection.

The line segment PP' has slope $\frac{3-9}{5-(-3)} = \frac{-3}{4}$ .

Thus the perpendicular bisector of PP', the line of reflection, has slope $\frac{4}{3}$ , since perpendicular lines have negative reciprocal slopes.

The line of reflection must also pass through the midpoint of PP', found at $(\frac{5+(-3)}{2}, \frac{3+9}{2}) = (1,6)$ .

We use the point-slope formula to find the equation of the line:

$y - y_1 = m (x-x_1)$

$y - 6 = \frac{4}{3} (x-1)$

With some rearrangement, we arrive at the equation 4x - 3y = -14

Thus, A + B + C = -13.

This is the same answer I got, but I was confused since the question says that A, B, and C do not have any common factors and clearly 4 and - 14 have a common factor of 2.